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talking with his whole body - especially his hands - to a spell-
bound group of students blocking the traffic. His house was
always open to his pupils, for Kronecker really liked people, and
Ms generous hospitality was one of the greatest satisfactions of
Ms life. Several of Ms students became eminent mathematicians,
but his 'school' was the whole world and he made no effort to
acquire an artificially large following.
The last is characteristic of Kronecker s own most startlingly
independent work. In an atmosphere of confident belief in the
soundness of analysis Kronecker assumed the unpopular role
of the philosophical doubter. Not many of the great mathema-
ticians have taken philosophy seriously; in fact the majority
seem to have regarded philosophical speculations with repug-
nance, and any epistemological doubt affecting the soundness
of their work has usually been ignored or impatiently brushed
With Kronecker it was different. The most original part of
his work, in which he was a true pioneer, was a natural out-
growth of his philosophical inclinations. His father, Werner,
Kummer, and his own wide reading in philosophical literature
had influenced him in the direction of a critical outlook on all
human knowledge, and when he contemplated mathematics
from this questioning point of view he did not spare it because
it happened to be the field of his own particular interest, but
infused it with an acid, beneficial scepticism. Although but
little of this found its way into print it annoyed some of his
contemporaries intensely and it has survived. The doubter did
not address himself to the living but, as he said, 'to those who
shall come after me'. To-day these followers have arrived, and
owing to their united efforts-although they often succeed only
in contradicting one another - we are beginning to get a clearer
insight into the nature and meaning of mathematics.
Weierstrass (Chapter 22) would have constructed mathe-
matical analysis on his conception of irrationals as defined by
infinite sequences of rationals. Kronecker not only disputes
Weierstrass; he would nullify Eudoxus. For him as for Pytha-
goras only the God-given integers 1,293, ... , 'exist': all the
rest is a futile attempt of mankind to improve on the Creator.